Ground State Eigenfunction Research Articles

T-Matrix Calculations of the Ground-State Energies of SolidHe3and SolidH2

The ground-state energies of solid ${\mathrm{He}}^{3}$ and ${\mathrm{H}}_{2}$ at various densities are calculated using a self-consistent method in the $t$-matrix formulation. The two-body equation of motion is solved by expanding the two-body wave function in terms of partial waves. The partial-wave expansion gives rise to a set of coupled differential equations which are solved numerically for the ground-state eigenfunctions. The calculations for ${\mathrm{He}}^{3}$ are done using three different two-body potentials, the Lennard-Jones potential, the Beck potential, and the Frost-Musulin potential. The calculations for ${\mathrm{H}}_{2}$ are done using the Mason-Rice two-body potential. A bcc structure is assumed for solid ${\mathrm{He}}^{3}$, while an fcc structure is assumed for solid ${\mathrm{H}}_{2}$. Exchange effects are neglected. Figures and tables are given which compare the present results with those of other authors.

A calculation of the energy and properties of the hydrogen molecule using the method of moments

The ground state energy and several properties of the hydrogen molecule are calculated using the method of moments. It is shown that when the expansion functions used on the right are capable of fitting the ground state eigenfunction very accurately, accurate results for the energy and properties are usually, but not always, obtained.

An iterative solution to the second order eigenvalue equation with periodic boundary conditions

A procedure for determining the ground state eigenvalue and eigenfunction of the Sturm-Liouville equation with periodic boundary conditions has been developed. This procedure is based on Picard's method, which, through the use of Chebyshev polynomials, provides for an efficient computational scheme.

Excitation Spectrum and Liquid-Structure Function of a Charged Bose Gas

The excitation spectrum of a charged Bose gas in the high-density limit is calculated to two orders in the expansion parameter ${r}_{s}^{\frac{3}{4}}$ by the method of correlated basis functions (CBF). To $O({r}_{s}^{\frac{3}{4}})$, the result differs from that obtained variationally, but agrees with that obtained by Ma and Woo via diagrammatic perturbation theory. The choice of the basis functions is, however, not unique. By choosing an alternate set of CBF which employs the ground-state eigenfunction as a correlating factor, we find it possible to extract from the excitation spectrum an expression for the liquid-structure function, also to two orders in ${r}_{s}^{\frac{3}{4}}$, which represents a new result. This new facet of CBF helps to demonstrate the flexibility of the method and the ease with which it can be applied to calculating properties of many-particle systems, model as well as realistic.

Paramagnetic resonance of Er3+ and Yb3+ in YAsO4

AbstractParamagnetic resonance of trivalent erbium and ytterbium in single crystals of yttrium arsenate has been observed at K‐band frequencies and temperatures of 4.2 and 1.6 K. Spin‐Hamiltonian parameters have been determined and the ground state eigenfunctions are discussed.

Energy Bands and the Optical Properties of LiCl

Using a recently developed local-orbital theory of the author's, the ground-state eigenfunctions of the LiCl crystal in the Hartree-Fock approximation have been obtained. Using these functions to form the crystal charge density, the one-electron energy bands have been computed in essentially the Hartree-Fock approximation using the mixed-basis technique. Various levels of approximation within the local-orbital approximation are investigated. Polarization corrections to the energy bands are included by the method of Fowler. Finally, using tight-binding interpolation for the valence levels and empirical pseudo-potentials for the conduction bands, the density of states for the valence and conduction states are obtained, and a joint density of states is computed. Extensive experimental comparisons are made. The principal results are the following: (1) The valence bands are very wide ($3p$ band is about 8 eV wide). (2) The soft x-ray spectrum can be interpreted in terms of the band structure. (3) Polarization corrections are significant.

On the calculation of Franck-Condon overlap integrals

A simple asymptotic equation is derived which describes the overlap integrals between the eigenfunctions of two harmonic oscillators, in particular between the ground state eigenfunction of one of the oscillators and the excited state eigenfunctions of the other oscillator. The equation is derived for large values of the quantum number n and it gives excellent agreement with the exact values if n is sufficiently large ( n > 10) and if the displacement between the two oscillators is not too small.

New Scaled Atoms-in-Molecules Theory for Predicting Diatomic Potential-Energy Curves III. Refined Applications to the X 1Σg+ and E 1Σg+ States of H2

Scaled atoms-in-molecules theory is applied to the ground X 1Σg+ and first excited E 1Σg+ states of H2. Five basis functions are included, corresponding to 1s H + 1s H, 1S H− + H+, 1s H + 2s H, 1s H + 2p0H, and 2p+ H + 2p− H. Variational scaling parameters are introduced into the A- and B-atom eigenfunctions comprising each basis function; the ground and first excited state energy roots of the secular determinant are independently minimized with respect to these scaling parameters. For computing interatomic contributions to the energy and overlap matrix elements, three different approximations to the 1S H− ground-state eigenfunction are used. Scaled atoms-in-molecules theory in this approximation yields an H2 ground-state energy which is 6.5 kcal/mole above the exact result. An E 1Σg+ excited-state curve is obtained with the proper double minimum; the predicted energy at the outer minimum is almost exact, and at the inner minimum it is about 7.5 kcal/mole too high.

Ligand-field theory for five-co-ordinate molecules. Part III. The magnetic properties of trigonal bipyramidal molecules with A ground terms

The energy levels for transition metal ions with the electronic configurations d2, d4, d7, and d9, under the combined action of spin–orbit coupling and a crystal field of trigonal bipyramidal geometry have been deduced and the magnetic properties, expected for such systems of levels, evaluated. For metal ions with these electronic configurations, and which have orbital singlet ground states in a crystal field of D3h symmetry, it is shown that the contributions to the ground-state eigenfunctions from higher lying, orbitally degenerate crystal-field states, affect only one of the principal molar susceptibilities, thus leading to anisotropic magnetic properties. The general formula for the magnetic moments of powder samples can be summarised as µeff=µ8·0. (1 –nλ/E) where n= 4 for the d2 and d7 configurations (assuming only the xE″ excited state from the xF state being mixed in) and n= 2 for the d4 and d9 configurations, and E is the energy between the A2′ and E″ or the A1′ and E″ crystal-field states respectively.The magnetic properties of some compounds of basically trigonal bipyramidal geometry are considered and using the experimental data for the complex VBr3(NMe3)2, a value for the spin–orbit coupling constant for the metal ion close to the free-ion value has been evaluated. In addition the magnetic properties of some complexes of C3v and lower symmetry are discussed in the light of the predicted behaviour, and for the d7 configuration the observed magnetic moments are larger than given by the model.

Paramagnetic resonance of Nd 3+, Dy 3+, Er 3+ and Yb 3+ in YVO 4

The paramagnetic resonance spectra of Nd 3+, Dy 3+, Er 3+ and Yb 3+ in YVO 4 were measured at X and Ka bands and at 4.2 and 1.8°K. Spin Hamiltonian parameters were determined and the ground state eigen-functions are discussed. The relaxation time of Er 3+ was measured at X band.

Ground-State Energy Eigenvalues and Eigenfunctions for an Electron in an Electric-Dipole Field

Ground-state energy eigenvalues and eigenfunctions are obtained by a variational method for an electron in the field of a finite, stationary, permanent electric dipole. The dipole moments studied cover the range from the minimum value required for binding (${D}_{min}=0.6393 e{a}_{0}$) to $D=400 e{a}_{0}$, where the system is equivalent to the hydrogen atom perturbed slightly by a distant stationary negative charge. The eigenvalues obtained agree with those reported by Wallis, Herman, and Milnes, who determined them by another method in the range $D=0.84e{a}_{0} \mathrm{to} 30e{a}_{0}$. The normalized eigenfunctions display the manner in which the electronic charge density changes from that of the hydrogen atom at very large $D$ to a flat distribution approaching that which is characteristic of a zero-energy continuum state as the minimum moment is approached from above. Optimized variational wave functions for different values of $D$ are presented for use in other calculations. Contour maps and profiles of electronic charge density are shown for a number of values of $D$. Mean values of the powers -1, 1, and 2 of the distances of the electron from the dipole charges are also calculated.

New Theory of Lattice Dynamics at 0°K

It has previously been shown that the optimum choice of harmonic Hamiltonian with which to approximate a crystal Hamiltonian is one is which the force constants are equal to the ground-state expectation value of the second derivative of the crystal potential, the expectation value being computed self-consistently with the ground-state eigenfunction of the harmonic Hamiltonian. It is shown here that the appearance of ground-state averages of various derivatives of the potential is due to an explicit or implicit expansion of the potential in a Hermite polynomial series, and that such an expansion is superior to the conventional Taylor-series expansion for anharmonic systems. In addition, it is shown that one can systematically treat very anharmonic systems by expanding the Hamiltonian in a set of polynomials orthogonalized with respect to weight function chosen to cut off the potential at short range, and that explicit incorporation of an easily optimized Gaussian factor in this weight function provides a computationally convenient way of introducing certain desirable features into the general expansion.

Momentum Eigenfunction for the Ground State of H2+

The nodes of the momentum eigenfunction for the ground state of the hydrogen molecule ion have been studied by calculating the momentum eigenfunction as a function of the momentum component Pz (z axis parallel to line of centers) with Px=Py=0. The existence of zeros in the ground-state eigenfunction is confirmed.

Properties of One-Dimensional Correlated Gaussian Wave Functions

The properties of a certain class of unsymmetrized one-dimensional correlated Gaussian wave functions---those which are ground-state eigenfunctions of some coupled harmonic-oscillator Hamiltonian---are investigated in detail. It is shown that a properly symmetrized wave function constructed from these may be used to calculate the expectation value ${E}_{0}$ of the Hamiltonian appropriate to a system of interacting one-dimensional atoms and that this energy is, to a high degree of accuracy, equal to the value obtained when the unsymmetrized wave function is used. A method is given by which correction terms to ${E}_{0}$ may be obtained. In addition, it is found that even though the number of particles be very large, the necessary multivariable integrals may be performed quite simply.

HARTREE–FOCK WAVE FUNCTIONS FOR EXCITED STATES: THE 2 1S STATE OF HELIUM

Hartree–Fock wave functions for the first-excited singlet states of several members of the helium isoelectronic sequence have been calculated. It is shown that unphysically low energies result if the excited-state function is not constrained to be orthogonal to the ground-state eigenfunction, and the excited-state orbitals are chosen orthogonal.If the excited-state orbitals are not chosen orthogonal, the effect of the overall constraint is small, raising the total energy very slightly; however, the effect of imposing both constraints simultaneously is appreciable. It is concluded that the most satisfactory approximation involves nonorthogonal excited-state orbitals, together with an overall orthogonality constraint towards the ground state.

Approximate Excited Eigenfunctions for Pr3+ and Tm3+

Approximate 5d, 6s, 6p, and 5f radial wave functions, which are necessary for any investigation of the effect of configuration interaction in rare-earth spectra, have been obtained for Pr3+ and Tm3+. The calculations were based on the ground-state eigenfunctions obtained by Ridley in an SCF calculation without exchange. 〈r2〉, 〈r4〉, Rk(4f4f, 4f6p), and Rk(4f4f, 4f5f) integrals have been calculated.

Perturbation method for low states of a many-particle boson system

A theoretical description of the ground state and low excited states of liquid He 4 is developed in terms of a set of correlated basis functions. We start from a simple correlated trial function ψ 0 suitable for an approximate description of the ground state under the assumption of a strong repulsive force when two particles approach closely. The function ψ 0 and a set of model functions ϕ n are used to construct a set, ψ n = ψ 0 ϕ n , of linearly independent correlated basis functions. Matrix elements of the identity and Hamiltonian operator are evaluated by systematic application of a generalized Kirkwood type superposition approximation. A normalized, orthogonal basis | e n > is constructed from linear combinations of the functions ψ m ; the associated matrix elements 〈 e n | H| e m 〉 vanish everywhere except on the three diagonals m = n, n ± 1. This result is a consequence of an appropriate choice of the model functions and use of the superposition approximation in evaluating the matrix elements. At this point is it proper to speak of a free phonon description. A final approximate diagonalization, neglecting phonon-phonon interaction, yields explicit formulas for the ground state energy and the momentum dependence of the phonon energy. The analogy with Bogoliubov's treatment of the boson system, using uncorrelated basis functions is very close as is also the relation to Feynman's theory of the excitation energies. A parallel analysis is successful with ψ 0 taken to be the correct ground-state eigenfunction. In this case the matrix elements of the phonon-phonon interaction can be expressed completely in terms of the elementary liquid structure function as given by the analysis of x-ray diffraction at low temperatures. The way is open to an accurate evaluation of the phonon energy as a function of momentum (the Landau curve) and a corresponding accurate evaluation of the thermodynamic properties of the liquid at low temperatures.

Hyperfine Splitting of the Lithium Ground State

The results of calculations of the hyperfine splitting of the lithium atom ground state are reported. These were done by the Hartree-Fock (HF), unrestricted Hartree-Fock (UHF) and projected UHF approximations to the ground-state eigenfunctions. The exchange polarization effect allowed by the UHF method yields a 34.9% increase in the hyperfine splitting compared to the HF method.

Quantum mechanics of the anharmonic oscillator

SummaryA method is presented for the accurate numerical treatment of molecular vibration problems in which the potential energy function is of the formThe treatment is carried through in detail for the case V(q) = Aq2 + Bq4, which, in most instances affords an adequate description of the potential. There is no restriction on the relative magnitudes of quadratic and quartic terms so that the method is equally applicable to the purely anharmonic oscillations occurring in types of ring bending (1), where V(q) = aq4.An approximate formula is derived for the energy levels in the general case. The case V = aq4 affords an illustration of the accurate treatment; the first five eigen-values and eigen-functions are computed and from them the associated transitionprobabilities. Numerical results are presented in a form convenient for further application. On comparison the approximate formula is found to yield results in error by approximately 1 %, the error decreasing for the higher levels for which the formula tends to a B.W.K. expression. In conclusion a ground-state eigen-function of simple analytical form is found to approximate remarkably well to the case of purely quartic vibrations.

On Tensor Forces and the Theory of Light Nuclei

The quadrupole moment of the deuteron indicates the existence of non-central tensor forces in nuclei which destroy the constancy of the total orbital angular momentum. With simple operational representations of the wave functions, the influence of two-body tensor forces on the ground state eigenfunctions of the light nuclei ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$ has been calculated. In ${\mathrm{H}}^{3}$ the tensor forces directly couple to the fundamental $^{2}S_{\frac{1}{2}}$ state a $^{4}D_{\frac{1}{2}}$ state, which in turn interacts with $^{2}P_{\frac{1}{2}}$ and $^{4}P_{\frac{1}{2}}$. To the fundamental $^{1}S_{0}$ state of ${\mathrm{He}}^{4}$ is admixed a $^{5}D_{0}$ state which is coupled by the tensor forces with $^{3}P_{0}$. All states consistent with the total angular momentum and parity conservation rules occur in the ground state eigenfunctions, and these nuclei therefore constitute the simplest examples of the complete break-down of spin and orbital angular momentum conservation laws. Rarita and Schwinger have satisfactorily accounted for the properties of the deuteron by including the tensor force in a simple interaction represented by a rectangular well potential. With this interaction to describe the forces between all pairs of nuclear particles, the binding energies of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$ have been estimated by a variation method The trial functions are of the form $^{2}S_{\frac{1}{2}}+^{4}D_{\frac{1}{2}}$ for ${\mathrm{H}}^{3}$ and $^{1}S_{0}+^{5}D_{0}$ for ${\mathrm{He}}^{4}$, with Gaussian radial functions. The calculations yield 32 and 50 percent of the binding energy for ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, respectively, while a similar test calculation for the deuteron gives 54 percent of the binding energy. The probability that these nuclei are in a $D$ state is found to be 4 percent for all three nuclei, in agreement with the exact deuteron computations. Improvement of the radial dependence of the trial functions increases the estimated binding energy of the deuteron to 76 percent of the known value but does not materially affect either the estimated binding energies of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$, or the amount of $D$ state admixture of the three nuclei. An analysis of the results shows that the tensor forces, which produce all the binding in the deuteron, are relatively ineffective in binding ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$. This apparently indicates that the assumption of ordinary and tensor forces of the same range is not adequate to represent the properties of ${\mathrm{H}}^{3}$ and ${\mathrm{He}}^{4}$.